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Polynomial and Rational Functions

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Polynomial and Rational
Functions
Lesson 2.3
Animated Cartoons
Note how
mathematics
are referenced
in the creation
of cartoons
Animated Cartoons
We need a way
to take a number
of points
and make
a smooth
curve
This lesson
studies
polynomials
Polynomials
General polynomial formula
P ( x ) пЂЅ a n x пЂ« a n пЂ­1 x
n
n пЂ­1
пЂ« ... пЂ« a1 x пЂ« a 0
• a0, a1, … ,an are constant coefficients
• n is the degree of the polynomial
• Standard form is for descending powers of x
• anxn is said to be the “leading term”
Note that each term is a power function
Family of Polynomials
Constant polynomial functions
• f(x) = a
Linear polynomial functions
• f(x) = m x + b
Quadratic polynomial functions
• f(x) = a x2 + b x + c
Family of Polynomials
Cubic polynomial functions
• f(x) = a x3 + b x2 + c x + d
• Degree 3 polynomial
Quartic polynomial functions
• f(x) = a x4 + b x3 + c x2+ d x + e
• Degree 4 polynomial
Properties of Polynomial Functions
If the degree is n then it will have at most
n – 1 turning points
•
•
•
End behavior
• Even degree
• Odd degree
or
or
Properties of Polynomial Functions
Even degree
• Leading coefficient positive
• Leading coefficient negative
Odd degree
• Leading coefficient positive
• Leading coefficient negative
Rational Function: Definition
Consider a function which is the quotient of
two polynomials
R ( x) пЂЅ
P ( x)
Q ( x)
Example:
r ( x) пЂЅ
2500 пЂ« 2 x
x
Both polynomials
Long Run Behavior
a n x пЂ« a n пЂ­1 x
n
Given
R ( x) пЂЅ
n пЂ­1
b m x пЂ« b m пЂ­1 x
m
m пЂ­1
пЂ« ... пЂ« a1 x пЂ« a 0
пЂ« ... пЂ« b1 x пЂ« b0
The long run (end) behavior is determined
by the quotient of the leading terms
• Leading term dominates for
large values of x for polynomial
• Leading terms dominate for
the quotient for extreme x
an x
n
bm x
m
Example
Given
3x пЂ« 8x
2
r ( x) пЂЅ
5x пЂ­ 2x пЂ«1
2
Graph on calculator
• Set window for -100 < x < 100, -5 < y < 5
Example
Note the value for a large x
3x
2
5x
2
How does this relate to the leading terms?
Try This One
Consider r ( x ) пЂЅ
5x
2x пЂ« 6
2
Which terms dominate as x gets large
What happens to
5x
2x
2
as x gets large?
Note:
• Degree of denominator > degree numerator
• Previous example they were equal
When Numerator Has Larger Degree
Try
2x пЂ« 6
2
r ( x) пЂЅ
5x
As x gets large, r(x) also gets large
But it is asymptotic to the line
yпЂЅ
2
5
x
Summarize
Given a rational function with
leading terms
When m = n
• Horizontal asymptote at
a
b
When m > n
• Horizontal asymptote at 0
When n – m = 1
• Diagonal asymptote
yпЂЅ
a
b
x
an x
n
bm x
m
Vertical Asymptotes
A vertical asymptote happens when the
function R(x) is not defined P ( x )
• This happens when the
denominator is zero
пЂЅ R ( x)
Q ( x)
Thus we look for the roots of the
2
denominator
x пЂ­9
r ( x) пЂЅ
x пЂ« 5x пЂ­ 6
2
Where does this happen for r(x)?
Vertical Asymptotes
Finding the roots of
the denominator
x пЂ« 5x пЂ­ 6 пЂЅ 0
2
( x пЂ« 6)( x пЂ­ 1) пЂЅ 0
x пЂЅ пЂ­ 6 or x пЂЅ 1
View the graph
to verify
x пЂ­9
2
r ( x) пЂЅ
x пЂ« 5x пЂ­ 6
2
Assignment
Lesson 2.3
Page 91
Exercises 3 – 59 EOO
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